Remember that we had a look at this equation earlier in the course:

(hat{sigma} = sqrt{frac{Sigma ((x – bar{x})^2)}{n-1}})

Up to now, we gained an understanding all of the pieces of information in this formula:

• (n) = the number of scores
• (x) = the data set we are working with
• (bar{x}) = average of the data set
• (Sigma) = the sum of the data set

We should now be able to put all of this knowledge together to calculate the Standard Deviation. Let’s have a look at this in small steps using our ages from the previous examples:

Starting with (x):
(x = 30, 21, 59, 45)

Now, let’s calculate (bar{x}):
(bar{x} = frac{30 + 21 + 59 + 45}{4})
(therefore bar{x} = 38.75)

Next, let’s look at how we can calculate (Sigma):
(Sigma ((x – bar{x})^2) = (30 – 38.75)^2 + (21 – 38.75)^2 + (59 – 38.75)^2 + (45 – 38.75)^2)
(therefore Sigma ((x – bar{x})^2) = (-8.75)^2 + (-17.75)^2 + (20.25)^2 + (6,25)^2)
(therefore Sigma ((x – bar{x})^2) = 76.5625 + 315.0625 + 410.0625 + 39.0625)
(therefore Sigma ((x – bar{x})^2) = 840.75)

And remember that we have four points of data, so:
(n = 4)

Let’s apply all of these into our formula:
(hat{sigma} = sqrt{frac{Sigma ((x – bar{x})^2)}{n-1}})
(therefore hat{sigma} = sqrt{frac{840.75}{4-1}})
(therefore hat{sigma} = sqrt{frac{840.75}{3}})
(therefore hat{sigma} = sqrt{280.25})
(therefore hat{sigma} = 16.74)

We shall discuss what standard deviation means later in this course. For now, we are learning how to apply formulas to statistics.

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